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Instructional
programs from prekindergarten through grade 12 should enable all students
to—
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Thinking mathematically involves looking for connections, and making connections builds mathematical understanding. Without connections, students must learn and remember too many isolated concepts and skills. With connections, they can build new understandings on previous knowledge. The important mathematical foci in the middle grades—rational numbers, proportionality, and linear relationships—are all intimately connected, so as middle-grades students encounter diverse new mathematical content, they have many opportunities to use and make connections.
This chapter on grades 6–8 mathematics contains numerous illustrations of
mathematical connections. Many of the formulas students develop and use
in the "Measurement" section draw on their knowledge of algebra, geometry,
and measurement. The kite example in the "Geometry" section engages students
in examining the perimeter and area of similar figures to investigate
proportional relationships. Several examples in the "Data Analysis" section
illustrate how gathering, representing, and analyzing data can help students
develop insights into other mathematical ideas, including variation and
change, probability, and ratio and proportion. The "cellular telephone"
problem in the "Algebra" section demonstrates how connections among various
forms of representation provide insights into patterns and regularities
in problem situations. Clearly, rich problem contexts involve connections
to other disciplines (e.g., science, social studies, art) as well as to
the real world and to the daily life experiences of middle-grades students.
Mathematics classes in the middle grades should continually provide opportunities for students to experience mathematics as a coherent whole through the curriculum used and the questions teachers and classmates ask. Students reveal the ways they are connecting ideas when they answer questions such as, What made you think of that? Why does that make sense? Where have we seen a problem like this before? How are these ideas related? Did anyone think about this in a different way? How does today's work relate to what we have done in earlier units of study? From these discussions, students can develop new connections and enhance their own understanding of mathematics by listening to » their classmates' thinking.
If curriculum and instruction focus on mathematics as a discipline of connected ideas, students learn to expect mathematical ideas to be related. Rich mathematical tasks prompt students to use and develop mathematical understandings and connections. Challenging problems encourage students to think about how familiar concepts and procedures can be applied in new situations. In classrooms where students are expected to reason mathematically and to communicate clearly about significant mathematical tasks, new ideas surface quite naturally as extensions of previously learned mathematics. With prompting from their teacher, students routinely ask themselves, "How is this problem like what I have done before? How is it different?"
Consider an expanded version of a summary (adapted from NCTM, Algebra Working Group [1998, p. 155]) of a lesson on ratio and proportion. The intent of this lesson was to begin developing students' understanding of methods for comparing ratios. The students had not previously been taught such methods, so the teacher wanted to uncover whether and how students could apply what they had already learned about number and ratio. The lesson was centered on the following task, which was adapted from Lappan et al. (1998, p. 27):
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Southwestern Middle School Band is hosting a concert. The seventh-grade class is in charge of refreshments. One of the items to be served is punch. The school cook has given the students four different recipes calling for sparkling water and cranberry juice.
The students worked on the first two questions in groups of two
or three. When the groups had finished, they came together as a
whole class to share and explain their answers. |
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The groups had attempted to figure out which recipe has the strongest cranberry flavor in different ways. Some examined the part-whole relationships of the number of cups of juice to the total number of cups in the recipe (these ratios are 2/5, 4/12, 3/8, 1/5 for recipes A–D, respectively). Others looked at the part-part ratios of juice to water (2/3, 4/8, 3/5, 1/4). Still others, failing to consider that the recipes, as given, make different amounts of punch, incorrectly » considered only the number of cups of juice in each recipe (2, 4, 3, 1). After questioning and challenging one another's solutions and comparing methods, the class decided to move on to the last question to see if they could resolve the differences in their answers. Each group was assigned to determine the amounts of juice and water needed for just one of the recipes. Below are four of the strategies the groups used to work through this part of the problem.
Group with recipe A
We figured out that each recipe would make 5 cups: 2 of juice and 3 of water.
So to make 120 cups, it would take 120 divided by 5, and that is
24, the number of recipes needed. Since we need 2 cups of juice
and 3 cups of water for one recipe, we need 2
Group with recipe B
We thought that 4 cups of juice and 8 cups of water is the same ratio as 1 cup of juice and 2 cups of water. We then thought about the 120 cups of punch as divided into three groups of 40 cups each: 40 + 40 + 40 = 120. We need 1 part juice, so that is 40 cups, and 2 parts water, so that is 80 cups. This makes 120 cups of punch, and you still have a ratio of 1 part juice to 2 parts water.
Group with recipe C
We tried to double the recipe, but that was not enough. So we added another batch and that still was not enough. So we just kept adding recipes and seeing how many total cups of punch we had. We kept up this pattern until we got 120 cups. So we had [a table like that shown in fig. 6.38]. That means we had 45 cups of juice and 75 cups of water.
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Later in the class discussion, this group noticed that they could have gone directly from 3/5 to 45/75 by multiplying the numerator and denominator by 15 because they needed 15 recipes.
Group with recipe D |
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We tried various numbers. First we tried 20 cups of juice. This means we needed
4 times as much water or 80 cups of water. But this was too small
because 20 + 80 is only 100. So we tried 30 cups of juice, so that
meant 30
After the groups had shared their approaches to the third question, the teacher continued the conversation by encouraging the class to talk about the similarities and differences among the strategies. |
The "making punch" problem had brought numerous mathematical ideas to the forefront: fractions, ratios, proportions, operations, magnitude, scaling, number sense, patterns, and so on. By bringing previously understood mathematical ideas or processes to bear on this problem, the students were developing understandings that laid a foundation for the later study of such topics as rates of change and linear relationships.
Since the task required the students to explain their strategies, all the students had an opportunity to enhance their understanding of ratios by listening to the others' different ideas. For example, the group with recipe D used a "guess and check" approach to solve the problem. The group with recipe C made a table and used the ideas of scaling ratios and adding iteratively in the same way that students find equivalent fractions. The groups with recipes A and B thought about comparing quantities and using ratios.
None of the students mentioned that the answers to the first two questions would
have been more obvious if they had solved the third problem first. For
each recipe, we can add the number of cups of cranberry juice to the number
of cups of water to determine how much punch one recipe makes. We divide
this number into 120 to determine the multiples—24, 10, 15, and 24,
respectively—of the ingredients that are needed. Because recipes
A–D use 2, 4, 3, and 1 cup of cranberry juice initially, they will
use 48, 40, 45, and 24 cups of cranberry juice, respectively, when multiplied
to serve 120 people. Clearly, recipe D has the weakest cranberry flavor
and recipe A has the strongest. This finding confirms the students' previous
answers and approaches.
It is sometimes quite effective to revisit a problem to help students connect familiar ideas to new concepts or skills. Indeed, the "making punch" problem has potential for connections to proportionality and linearity. For instance, students could make a graph, plotting values » from the first and third rows of the table in figure 6.38. These points lie on a line. If y represents the total number of cups of punch and x the number of cups of juice, then the line has equation y = (8/3)x (see fig. 6.39). To answer question 3—which asks how much juice and how much water are needed for 120 cups of punch—students can substitute 120 for y in this equation and compute x to find the number of cups of juice. They would subtract this number from 120 to find the amount of sparkling water needed. This equation works equally well for finding how much juice is needed for any other quantity of punch, so they see the power of expressing a relationship in general terms. Here the slope of the line is the ratio of the amount of punch to the amount of juice. Of course, having used this method to answer question 3 for all recipes, students can easily answer the other two questions. A teacher might also revisit the "making punch" problem to assess students' understanding of tabular, graphical, and symbolic representations for linear relationships.
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In many schools, teachers are interested in fostering interdisciplinary studies. The mathematics teacher may work with teachers of other subjects to develop integrated units of study. For example, middle-grades science classes might study populations of wildlife such as deer, fish, eagles, or sharks (see Curcio and Bezuk [1994]). If students will be expected to use sampling techniques in science class to determine the population of a species, it is important that the mathematics and science teachers discuss students' understanding of different sampling techniques and of the idea of randomness. It is also important that science teachers understand that students are likely to use scaling and equivalent ratios to estimate the total population » rather than cross multiplication and formal algebraic symbol manipulation to find their solutions.
In the same spirit, mathematics teachers can build on and connect to disciplines other than science and social studies. For example, language arts teachers can describe the strategies they teach for writing convincing arguments. The mathematics teachers may then be able to help students use the strategies when appropriate in formulating mathematical arguments. They may also be better able to help students recognize and analyze forms of argumentation and justification that are peculiar to mathematics. Again, students benefit from teachers' efforts to understand how other subjects are taught and to make connections between the subjects explicit.
Conversations about students' experiences, understandings, and familiarity with procedures give teachers of other subjects an opportunity to learn about elements of the mathematics curriculum, such as algorithms and the level of abstract symbol manipulation that students might use. Without such conversations, those who are not mathematics teachers may expect students to understand and use procedures that are not part of their repertoire or teachers may fail to build on ideas with which students are already conversant. Students may miss an opportunity to apply and extend their reasoning skills or to see that mathematical ideas can be used in other disciplines. This is not to imply that merely applying mathematics in science, social studies, or any other discipline constitutes a sufficient middle-grades mathematics curriculum. The point is that interdisciplinary experiences serve as ways to revisit mathematical ideas and they help students see the usefulness of mathematics both in school and at home. If all the middle-grades teachers in a school do their best to connect content areas, mathematics and other disciplines will be seen as permeating life and not as just existing in isolation.
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24 = 48 cups of juice
and 3
